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# Statics and Dynamics MEEN 2130

UNT

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This 11 page Class Notes was uploaded by Ofelia Krajcik on Sunday October 25, 2015. The Class Notes belongs to MEEN 2130 at University of North Texas taught by Zhi-Gang Feng in Fall. Since its upload, it has received 51 views. For similar materials see /class/229134/meen-2130-university-of-north-texas in Mechanical Engineering at University of North Texas.

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Date Created: 10/25/15

MEEN2130 STATICS AND DYNAMICS 2003 Chapter 16 Planar Kinematics ofa Rigid Body Rigid Body 7 i 7 Rigid Body Panicle Muluple Particles J V ELisnc Body Rigid Body Motion I Translation dun39ng die motion and acceleration Remarks 1 ran Wah nn translation depending on me pad 3 panieie Lakes 2 39 39 nanaaii nie nioiion along pain 2 is die curvilinear translation 1 Dept of Meeh Energy and Eng Univ of Norm Texas MEEN2130 STATICS AND DYNAMICS 2008 Rigid Body Motion II Rotation about a xed axis Pure Rotation In this case a particle on the body move along a circular path except the ones on the line of the xed axis which remain stationary xed a view from 3D perspective view in 2D Angular motion of the rigid body Angular Position 6 Angular displacement d5 Angular velocity magnitude a direction along the rotation axis as 15 Angular acceleration magnitude 1 Iii f direction along the axis Di erential relationship 19 d da adt adQ uda Integral relationship amfotocdt7 9QOftmdt7 m2m 2fgoad9 In particular if angular acceleration is a constant a 15 then the above integrals yielda no act 9 90 not act2 m2 mg 20559 90 Motion of a point P attached to the rigid body in pure rotation Velocity Vector Scalar 1 an along tangential direction l I Dept of Mech Energy and Eng Univ of North Texas MEEN2130 STATICS AND DYNAMICS 2003 Accelera n Vector Scalar a m 12 My Remarks L at J rigid bodyhave different velocities and accelerations 2 body 1 1 Dept of Meeh Energy and Eng Univ of North Texas MEEN2130 STATICS AND DYNAMICS 2008 Rigid Body Motion 111 General Planar Motion A body undergoes both translation and rotation It can be Viewed as a rotation wrt a translating base point axis L The general plane motion can be considered as a combination of translation and rotation 39 I translation rotation wrt A general plane motion Relativevelocity acceleration analysis for general planar motion Consider the motion of base pointA and another arbitrary point B in a fixed axis xy FB 2 FA FBA 3953 1 724 5314 33 2 EA EBA But the relative motion of B is caused by the rotation of line segment rBA wrt the base pointA more precisely the axis passing through A hence 53 13A aixFBA a3 3A aBA 3A 3 x FBA w2FBA 1 I Dept of Mech Energy and Eng Univ of North Texas MEEN2130 STATICS AND DYNAMICS 2008 Instantaneous center of zero velocity IC At each instant there is an Instantaneous Center onero Velocity IC The motion at that instant can be viewed as a pure rotation wrt the 1C the velocity only the velocity can be computed use the formula for pure rotation Location ofthe IC owin V and V 2 14 10 mm A Case 2 Knowing V4 and V I I IC a2mgt4IOngAL Remarks 1 For a true general planar motion there exists no fixed point zero velocity and zero acceleration 2 The acceleration of IC itself will not be zero the acceleration of any other point still has to be computed by using the general formula 3 The IC can be outside of the rigid body on the extension of the body The pure rotation can be regarded as a special case in which the IC is xed and the translation is a special case in which the IC is at in nity 1 I Dept of Mech Energy and Eng Univ of North Texas MEEN2130 STATICS AND DYNAMICS 2008 Procedure to Solve Rigid Body Kinematics Step 1 Draw a kinematical diagram or each rigid body 11 Establish coordinate system set the direction of x y and z 12 Select the instantaneous rotating axis for a general plane motion consider picking the center of mass pivot or a known velocity point as the base point we denote it with A Indicate the translating velocity 13A and acceleration 07A of the base point 13 Indicate the angular velocity and acceleration of the rigid body for unknown one assume it in positive sense counterclockwise 14 Locate the interest point indicate its velocity 133 acceleration 63 and indicate relative position vector PBA as well Step 2 Express the related vectors as Cartesian vectors 21 In particular write FBA 13A 01A 133 63 Q d as Cartesian vectors if they are known 22 If only the direction of a vector is known try to express this vector with one unknown variable 23 You may also have to apply any other proper constraints such as sliding along a surface connected to another body etc to reduce the unknowns used in describing the above vectors Step 3 Applu the relative quothnih39 accelerau uu 39 39 31 Use g 13A 6 X PBA for velocity analysis and 63 01A BugA 07A d X FBA 1273 A for acceleration analysis 32 While performing vector cross use the identities I X Y j and I X j Y 33 Generally the acceleration analysis needs the results from the velocity analysis Step 4 Solve the to obtain the variables 41 After performing the vector cross products turn the vector equation into two algebra equations by comparing both Yand f components 42 If there are more than three unknowns appeared in the two algebra equations you need nd an additional equation for them likely from the known direction of a vector otherwise solve the unknowns 43 If any solution yields a negative answer it simply indicates the sense of direction of the vector is opposite to the assumption Ste 39 I necessa re eat above ste s or more oints or or anotherri id bod 1 Dept of Mech Energy and Eng Univ of North Texas Examples 1 What type of motion for each moving component below 73 2 A box is sliding along a smooth surface at 5ms What is the velocity ofA relative to B A 34M 3 A rigid bar is rotating about a xed axis 0 with angular displacement 9 2 32 The distance from axis 0 to the end point P is R 1m Please find the linear velocity and acceleration of point P at t 1s 4 Abar is rotating around an axis xed on a blockA with a constant angular velocity u 2 10 rads R1m What is the relative velocity and acceleration of point C compare to point B if a the blockA is not moving 1 the block A moving with velocity 1 ms and acceleration of 02 ms2 9quot MEEN2130 STATICS AND DYNAMICS I 2008 Given a linkage system as shown DAB 10 rads and aAB1 rads2 both counter clockwise Find the angular velocity mm and the angular acceleration 1CD 9 The 3m long ladder AB slides along a corner At the instant 0 30 and the lower end A is moving to the right at a constant speed of 1ms Determine the velocity and the acceleration at point B together with the angular velocity and angular acceleration of the ladder 1 I Dept of Mech Energy and Eng Univ of North Texas MEENZ 130 STATICS AND DYNAMICS 2008 Chapter 17 Planar Kinetics of A Rigid Body Force and Acceleration Moment otInertiu otBody otMuss 111 about zAxis Iz fm r2 dm or 10 fm r2 dm in 3d perspective Perpendicular Axis Theorem As shown above Iz Ix Iy ParallelAxis Theorem 10 16 1erZ Radius of gyrution k kJI or ImkZ m Remarks The moment of inertia about an axis can be computed by dividing the body into several subbodies in simple geometry shape and sum all the moments of inertia of these simple shapes wrt the same axis This also works for composite body Chap 17 I Dept of Mech Energy and Eng Univ of North Texas MEENZ 130 STATICS AND DYNAMICS I 2008 M ament u Inertia about 0 urA Cu usite Bu lg 205 de A Freerbodv litlgmm In Vector form Kinetic diugrmn Z mag ME 2 x mar Had 22am left side quot 393 uia Inull In scalar form 2 Fa max 2 F3 magy 2 Ma 1511 7 Dr EM Yam lah Xemaay 1511 3 2617201 Remarks mama yp1ahe L 39 h rempoheht in zrh39reetioh unit Vector k 2 The force equations rah be wn39tten in other coordinates eg natural orpolar coordinate to faeihtate the calculation 3 L c hr ever in L a point other than G say point P Chap 17 Dept of Mech Energy and Eng Univ of North Texas MEEN2130 STATICS AND DYNAMICS 2008 4 The 11 can be computed using the parallel theorem 5 The kinetic moment contains the moment by inertial force mag and the Is a Special case 1 Pure Translation 2 1quot moi G or two scalar equations in selected coordinate 2 MG 0 Special case 11 Pure Rotation wrt A Fixed PointgAxis 0 2 Ft ma6t mars0 ZFn ma6n mMTG0 2M0 1001 W Draw the FBDs Established a proper coordinate system You can set up different coordinate for each different rigid body Find the base point rotating axis the center of the mass or a pivot point N Apply the equation of motion in the chosen coordinate Apply kinematic analysis Apply any other geometric constraints Solve the equations to get the unknown variables NP WPS Chap 17 I Dept of Mech Energy and Eng Univ of North Texas

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